Questions tagged [prime-numbers]
Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.
2,020
questions
0
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0
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104
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Variants of Nicholson's inequalities for prime numbers, involving the Lambert $W$ function
The purpose of this post is ask about two closely related/inspired conjectures from inequalities due to Nicholson (see [1]) and Visser [2]. If my reasonings are right should be stronger versions of ...
0
votes
0
answers
128
views
A quadratic trinomial that generates only prime numbers of the form $4m+1$
It is known that Euler's polynomials $\,n^2+n+p\,$ ($p\,$ prime) represent a prime for $\,n=0,\,...,\,p-2\,$ if and only if the field $\,Q (\sqrt{1-4p})\,$ has class number $\,h=1$.
The best ...
6
votes
1
answer
227
views
Is there a connection between the average 'compositeness' of a rational number and $\phi$ (golden ratio)?
Let $n\in N$, where $n = p_{1}^{k_{1}}p_{2}^{k_{2}}...p_{m}^{k_{m}}$ for $p_{i}$ prime.
Define the 'density' of $n$ as:
$d(n) = \frac{(p_{1}+1)^{k_{1}}(p_{2}+1)^{k_{2}}...(p_{m}+1)^{k_{m}}}{n}$
...
0
votes
1
answer
159
views
does the ratio of the count of rational numbers on an $n\times n$ grid to $n^2$, converge as $n$ tends to infinity [closed]
Suppose we order the rational numbers using the diagonal method (used to prove they are countable) using an $n\times n$ grid. Now suppose we count the distinct rational numbers (those points on the ...
0
votes
1
answer
237
views
What can this $\int_{0}^{t} (\pi(x)-Li(x)) dx$ tell us about primes distribution?
Many papers I have read which are related to primes distribution only it discussed sign and refinement Bounds of $\pi(x)-Li (x)$ with $\pi(x)$ is a prime counting function and $Li (x)$ is the ...
1
vote
1
answer
188
views
Resolution of an inequality on integers
I’m trying to resolve respect to $k$ the following inequality,
$$
k\left(\log k +\log \log k-\alpha+O\left(\frac{\log \log k}{\log k}\right)\right)\geq x,
$$
in order to obtain, under the condition $...
1
vote
0
answers
142
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About the distribution of Fibonacci numbers that are primes
Let's consider the Fibonacci sequence, that is the sequence of naturals defined by:
$F_1=F_2=1$
$F_{n+1}=F_{n}+F_{n-1}$
It is an open problem whether the sequence contains an infinite number of ...
4
votes
0
answers
143
views
Moments of the prime counting function given the moments of the second Chebyshev function
I have read this article (Montgomery and Soundararajan: Primes in short intervals. http://arxiv.org/abs/math/0409258 ). In the second page of the article, it is stated that the mean and variance of $\...
1
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0
answers
175
views
Given a prime $\,p\ne3$, is it always possible to find another prime q such that $\,\phi(q)=\phi(p\cdot2^{2n+1})\,$ for some $n\in\mathbb{N}$?
Given a prime $\,p\ne3$, is it always possible to find another prime q such
that $\,\phi(q)=\phi(p\cdot2^{2n+1})\,$ for some $n\in\mathbb{N}$ ($\,\phi\,$ is the Euler's totient function)?
Some ...
3
votes
1
answer
310
views
Why is this sequence a good prime-generator?
For $n \in \mathbb N$ we can observe the $n$ remainders $b_1,...,b_n$ by writing $n$ as $n=a_k \cdot k+b_k$ for $1 \leq k \leq n$.
Because of the familiar division-with-remainder theorem we have $0 \...
11
votes
1
answer
421
views
How many numbers $\le x$ can be factorized into three numbers which form the sides of a triangle?
Note: Posting in MO since it was unanswered in MSE
Definition: We say that a natural number $n$ has triangular divisors if it has at least one triplet of divisors $n = d_1d_2d_3, 1 \le d_1 \le d_2 \...
-1
votes
1
answer
281
views
Is it possible to determine whether the sequence $\,a_0=p,\;a_{n+1}=(a_n-2)\cdot a_n+2\,$ will reach another prime number?
Given a prime $\,p\,$ let's consider the following sequence:
$a_0=p$
$a_{n+1}=(a_n-2)\cdot a_n+2$
Is it possible to determine whether the sequence $\,a_n\,$ will reach, sooner or later, another ...
1
vote
0
answers
277
views
Prime numbers in this region
Let $q \geq 5$ be a prime number, and consider : $N_q = \displaystyle{\small \prod_{\substack{p \leq q \\ \text{p prime}}} {\normalsize p}}$
Using Chinese remainder theorem we can show that :
$$\#\{(...
0
votes
0
answers
141
views
Given $\,m=\prod_k {p_k}^{\alpha_k}\,$ and the function $\,g(m)=\sum_k \alpha_k(p_k-1)^2$, find all solutions of the equation $\,g(2n)=n$
Let's consider the unique decomposition of a natural number $\,m\,$ into its prime factors:
$$\prod_k {p_k}^{\alpha_k}$$
Then, let's define the following arithmetic function (completely additive) $\,g:...
3
votes
1
answer
266
views
Solutions in primes of the equation $\,3p^2+q^2=r^2+3$
Let's consider the Diophantine equation $\,3p^2+q^2=r^2+3$.
Actually, I am interested only in the solutions represented by sets $\,(p,q,r)\,$ of prime numbers.
It's easy to prove that if $\,(p,q)\,$ ...
13
votes
1
answer
1k
views
About the number of primes which are the sum of 3 consecutive primes (OEIS A034962)
I made some numerical simulations about the number of primes which are the sum of 3 consecutive primes (OEIS A034962), that is for instance:
$$5+7+11=23$$
$$7+11+13=31$$
$$11+13+17=41$$
$$17+19+23=59$$...
15
votes
1
answer
1k
views
Greatest prime factor of n and n+1
For a positive integer $n$ we denote its largest prime factor by $\operatorname{gpf}(n)$. Let's call a pair of distinct primes $(p,q)$ $\textbf{nice}$ if there are no natural numbers $n$ such that $\...
1
vote
1
answer
116
views
Sequences of positive integers $(a_{k})_{k \in \omega}$ that only give finitely many zeros modulo $p_{k}$ in total for all polynomials
Let $(a_{k})_{k \in \omega}$ be a sequence of positive integers such that $a_{k} < p_{k}$, $a_{k} \leq a_{k+1}$ and $\lim_{k \rightarrow \infty} a_{k}=\infty$ where $p_{k}$ is the k-th prime ...
2
votes
0
answers
170
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Funny questions about Moebius Function
I need to firstly claim that my research is not about number theory, however, I am pretty interested in it, especially funny questions in number theory, e.g. Kollatz Conjecture. Three years ago, I ...
0
votes
0
answers
77
views
Construction of (general class of) function(s), which sieves out primes, w.r.t. given conditions:
Consider the function $F(x)$ defined in following manner:
$F(n)$ is finite (likely $F(x)\in[0,1]$) if $n$ is prime and zero otherwise:
It has to satisfy following conditions:
(1) $F(x)$ is ...
2
votes
1
answer
321
views
Can someone explain some of the steps in this paper clearly?
I'm having trouble understanding the steps this paper makes to come to the conclusion $p_{f}(d) \sim e^d\sqrt{d}$
Marek Wolf, First occurrence of a given gap between consecutive primes, preprint, ...
0
votes
1
answer
352
views
A sufficient condition for a set of primes to be the set of reducibility of an integer polynomial
Let $P$ be the set of all positive primes. Let $S$ an arbitrary infinite subset of $P$ satisfying the following assumption: there exists a finite Galois extension $K$ of $\mathbb{Q}$ and a conjugacy ...
2
votes
1
answer
525
views
Sets of primes with a given Frobenius conjugacy class
Let $a$ and $b$ be two relatively prime positive integers. Denote by $S$ the set of all positive primes of the form $a+bn$ (where $n$ is an integer, possibly zero or negative). Let $S'$ be any set of ...
4
votes
1
answer
313
views
Estimating certain short Kloosterman sums
Recall that for the classical Kloosterman sum
$$ K(a,b,p^t):= \sum_{x \in (\mathbb{Z}/ p^t \mathbb{Z})^* } \psi \left(\frac{ax+bx^{-1}}{p^t} \right),$$
where $\psi(x)=e^{2\pi ix}$, $a,b,t$ are natural ...
2
votes
1
answer
161
views
Numerical estimates for a function relating to twin primes :
Consider the following function :
$$F(s)= \sum_{\text{$p,\ p+2$ are primes}} \left({\frac{1}{p^s}}+{\frac{1}{(p+2)^s}}\right).$$
Brun's theorem tells us that $F(1)$ is finite.
We are looking for ...
1
vote
2
answers
141
views
Is the abscissa of convergence of $s\mapsto\sum_{n>0}(ng_{n}/2)^{-s}$ known?
The famous Polignac conjecture posits that the $n$-th prime gap $g_{n}:=p_{n+1}-p_{n}$ attains all even positive integral values infinitely many times, which implies $\displaystyle{\zeta_{Pol}:=s\...
4
votes
1
answer
819
views
Primality test similar to the AKS test
Let us define polynomials $P_n^{(a)}(x)$ as follows :
$P_n^{(a)}(x)=\left(\frac{1}{2}\right)\cdot\left(\left(x-\sqrt{x^2+a}\right)^n+\left(x+\sqrt{x^2+a}\right)^n\right)$
We can define these ...
2
votes
0
answers
156
views
Questions about a certain sequence of naturals generated by primorials
I'm working on the following sequence of naturals (which is NOT listed in OEIS)
$$3,5,11,17,23,29,59,89,119,149,179,209,419,629,839,1049,1259,1469,1679,...$$
whose elements are generated this way
$$3=(...
-1
votes
1
answer
242
views
A conjecture about an inequality that involve Ramanujan primes
In this post we denote for integers $n\geq 1$ the $n$-th Ramanujan prime as $R_n$ (thus the sequence A104272 from the On-Line Encyclopedia of Integer Sequences), I add a conjecture that I think can be ...
4
votes
2
answers
1k
views
Calculating the infinite product from the Hardy-Littlewood Conjecture F
The Hardy-Littlewood Conjecture F [1] involves the infinite product
$$\prod\left(1-\frac{1}{\varpi-1}\left(\frac D\varpi\right)\right)$$
where $\varpi$ ranges over the odd primes and $\left(\frac D\...
2
votes
0
answers
119
views
On the set $\{n>0:\ n\ \text{is a quadratic nonresidue modulo the}\ n\text{th prime}\}$
Let $S$ denote the set of positive integers $n$ with $n$ a quadratic nonresidue modulo the $n$th prime $p_n$. The first 20 elements of $S$ are
$$2,\, 3,\, 6,\, 7,\, 8,\, 10,\, 11,\, 13,\, 15,\, 18,\, ...
0
votes
1
answer
132
views
A density zero set of primes dividing the values of a non-constant integer polynomial
For a given $P\in \mathbb{Z}[x]$ call a positive prime $p$ good if there exists $n\in \mathbb{Z}$ such that $p$ divides $P(n)$. Does there exist a non-constant $P$ such that the set of good primes has ...
5
votes
1
answer
164
views
Representation of primes of the form $4m+3$ with double radicals
Let $\,q\,$ be a prime of the form $\,4\, m_q+3$.
I ask if it is always possible to find two primes $\,p_1$ and $\,p_2$ of the form $\,4\, m_p+1$ such that
$$q=\sqrt{p_1+\sqrt{p_2+q}}$$
E.g.
$$3=\...
2
votes
1
answer
289
views
Convergence of series $\sum_{k=1}^{\infty}\frac{p_{k+1}-p_k}{(p_{k+1}+p_k)^\alpha}$
I ask if the series
$$\sum_{k=1}^{\infty}\frac{p_{k+1}-p_k}{(p_{k+1}+p_k)^\alpha}$$
where $p_k$ stands for the prime of index $k$,
has the same properties of convergence of the series $$\sum_{k=1}^{\...
6
votes
0
answers
197
views
Smooth integers with lower bound on $\omega(n)$
Define $(b,c)$-smooth integers to be integers having all prime factors bigger than $c$ and smaller than $b$.
Probability a number is $(b,1)$-smooth is governed by the Dickman function while ...
-3
votes
1
answer
178
views
Asymptotic behavior of $\sum_{k=1}^{n}\frac{p_{k+1}}{p_{k+1}-p_k}$
I refer to my previous question Asymptotic behavior of a certain sum of ratios of consecutives primes.
We can split the sum
$$\sum_{k=1}^{n}\frac{p_{k+1}+p_k}{p_{k+1}-p_k}$$
where $p_k$ stands for the ...
10
votes
1
answer
465
views
Asymptotic behavior of a certain sum of ratios of consecutives primes
I am looking for the asymptotic growth of the following sum
$$\sum_{k=1}^{n}\frac{p_{k+1}+p_k}{p_{k+1}-p_k}$$
where $p_k$ stands for the prime of index $k$.
Manual computations show, for small values ...
6
votes
3
answers
821
views
Is it possible to multiply two series to get as a result all composite numbers?
I was toying with the following problem:
Is it possible to find two infinite integer sequences $(a_n), (b_n)>0$ such that $\sum_{n=1}^{\infty}\frac{1}{(a_n)^s}\cdot \sum_{n=1}^{\infty}\frac{1}{(b_n)...
7
votes
1
answer
220
views
The asymptotic of $|\{1\leq n\leq x|\gcd(n,S(n))=1\}|$, with $S(n)$ the sum of remainders, and get idea for other miscellany problem
Let $n\geq 1$ be an integer. In this post we denote the sum of remainders function as $$S(n)=\sum_{k=1}^n n \bmod k,$$ for example $S(1)=S(2)=0+0$ and $S(5)=0+1+2+1+0=4$. In the literature there are ...
4
votes
0
answers
141
views
Can this number be interpreted as a fractal dimension?
Under Goldbach's conjecture, let's denote for a large enough integer $n$ by $r_{0}(n)$ the quantity $\inf\{r>0,(n-r,n+r)\in\mathbb{P}^2\}$ and by $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n+r_{0}(n))$.
Let's ...
0
votes
0
answers
64
views
Can power of a prime number be approximated by product of powers of two adjacent prime numbers?
With prime numbers $a < b < c$ and no primes exist in ranges $(a, b)$ and $(b, c)$, is it possible that there exists positive integers $x$, $y$, $z$ such that $|a^x c^z-b^y|=2$?
12
votes
0
answers
2k
views
Would the following conjectures imply $\lim\inf_{n\to\infty}p_{n+k}-p_{n}=O(k\log k)$?
Assume Goldbach's conjecture. Then for every $n\ge 2$ there exists at least one non-negative integer $r\le n-2$ such that both $n+r$ and $n-r$ are primes. Let's write $r_{0}(n):=\inf\{r\le n-2, (n-r,n+...
5
votes
0
answers
136
views
Is finding positive integer solutions of $\zeta(a/b) = c$ equivalent to deciding the rationality of $\gamma$?
This question requires little bit of explanation of the background hence it is a bit lengthy. Note: The question was initially posted in MSE but did not get answers hence posting in MO.
For every ...
4
votes
0
answers
139
views
Is it true that $|\{k^{k+1}+(k+1)^k\pmod p:\ k=0,\ldots,p-1\}|=(1-e^{-1})p+O(\sqrt{p})\ ?$
For each prime $p$, let us define
$$w_p:=|\{k^{k+1}+(k+1)^k\pmod p:\ k=0,\ldots,p-1\}|,$$
where $a\pmod p$ denotes the residue class $a+p\mathbb Z$.
Based on my computation, I conjecture that
$$w_p=...
4
votes
0
answers
163
views
Primitive roots modulo primes related to Fibonacci numbers or Lucas numbers
The Fibonacci numbers $F_0,F_1,F_2,\ldots$ and the Lucas numbers $L_0,L_1,L_2,\ldots$ are given by
$$F_0=0,\ F_1=1,\ \text{and}\ F_{n+1}=F_n+F_{n-1}\ (n=1,2,3,\ldots)$$
and
$$L_0=2,\ L_1=1,\ \text{...
12
votes
2
answers
1k
views
$Ax^2 + By^3$ representing infinitely many primes
Are there any known results of the form
there are infinitely many primes of the form $Ax^2 + By^3$
for integers $A$, $B$?
Assuming there are currently no known results of this form, what is the ...
6
votes
0
answers
421
views
Average value of $\prod_{p|d}{p-1\over p-2}$ for $d=nq$, $n\in{\mathbb N}$, with $p$ odd prime
$\newcommand{\mean}{\mathop{\mathrm{mean}}}$
Define
$$
S(d) = \prod_{p|d\atop p>2}{p-1\over p-2}.
$$
Bombieri and Davenport (1966) proved that
$$
\mean\limits_{d\in{\mathbb N}} S(d) =
\mean\...
5
votes
0
answers
612
views
is there a link with the probabilistic model for prime numbers?
Let $x \in \mathbb{R}_+$ and $k \in \mathbb{N}^{*}$.
Let :
$$\mathcal{A}(x)=\#\{(a_1, a_2, \ldots, a_k) \in \mathbb{P}^k \mid (a_1, a_2, \ldots, a_k \text{ verifying some properties}) \, , a_k \...
4
votes
1
answer
875
views
Least Prime Factors: found a counting formula for a given range -- what is the standard approach?
Hi Everyone,
I am a math amateur who for the past year has been working on better understanding Bertrand's Postulate, the Ramanujan Primes, and the recent expansion of Bertrand's Postulate (always a ...
1
vote
1
answer
213
views
Prime analogue of Champernowne's constant
Are any non-trivial properties known about the constant 0.2357111317192329... that is obtained by catenating the digits of sequence of prime numbers in base 10 or in other bases, especially whether it ...